Derivative

Definitions: Derivative

Derivative

Adjective

1. Resulting from or employing derivation; "a derivative process"; "a highly derivative prose style".

Noun

1. The result of mathematical differentiation; the instantaneous change of one quantity relative to another; df(x)/dx.

2. A financial instrument whose value is based on another security.

3. (linguistics) a word that is derived from another word; "`electricity' is a derivative of `electric'".

Date "derivative" was first used in popular English literature: sometime before 1611. (references)

Specialty Definitions: Derivative

Domain	Definitions
Finance	A contract the value of which changes in concert with the price movements in a related or underlying commodity or financial instrument. The term covers standardised, exchange-traded futures and options, as well as over-the counter swaps, options, and other customised instruments ; contracts such as options and futures whose price is derived from the price of the underlying financial asset. Source: European Union. (references)
Language	An orthographic word in which one of the components is an affix. Source: European Union. (references)
Source: compiled by the editor from various references; see credits.

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Specialty Definition: Derivative

(From Wikipedia, the free Encyclopedia)

A derivative is an object that is based on, or created from, a basic or primary source. This meaning is particularly important in linguistics and etymology, where a derivative is a word that is formed from a more basic word. Similarly in chemistry a derivative is a compound that is formed from a similar compound. In finance, derivative is the common short form for derivative security.

In mathematics, the derivative of a function is one of the two central concepts of calculus. The inverse of a derivative is called the antiderivative, or indefinite integral.

The derivative of a function at a certain point is a measure of the rate at which that function is changing as an argument undergoes change. A derivative is the computation of the instantaneous slopes of f(x) at every point x. This corresponds to the slopes of the tangents to the graph of said function at said point; the slopes of such tangents can be approximated by a secant. Derivatives can also be used to compute concavity.

Functions do not have derivatives at points where they have either a vertical tangent or a discontinuity.

Differentiation and Differentiability

Differentiation can be used to determine the change which something undergoes as a result of something else changing, if a mathematical relationship between two objects has been determined. The derivative of f(x) is written in several possible ways: f'(x) (pronounced f prime of x), d/dx[f(x)] (pronounced d by d x of f of x), df/dx (pronounced d f by d x), or D_x[f] (pronounced d sub x of f). The last three symbolisms are useful in considering differentiation as an operator, and these symbolisms are known as the differential operator.

A function is differentiable at a point x if its derivative exists at this point; a function is differentiable in an interval if a derivative exists for every x within the interval. If a function is not continuous at c, then there is no slope and the function is therefore not differentiable at c; however, even if a function is continuous at c, it may not be differentiable.

Newton's Difference Quotient

Derivatives are defined by taking the limit of a secant slope, as its two points of intersection (with f(x)) converge; the secant approaches a tangent. This is expressed by Newton's difference quotient; where h is Δx (the distance between the x-coordinates of the secant's points of intersection):

Suppose one wishes to find the derivative of a suitable function, f(x), at x. If one increases x by some small amount, Δx, one can calculate f(x + Δx). An approximation to the slope of the tangent to the curve is given by (f(x + Δx) - f(x)) / Δx, which is to say it is the change in f divided by the change in x. The smaller Δx is, the better the approximation is. From here on, Δx will be referred to as h. Mathematically, we define the derivative to be the limit of this ratio, as h tends to zero.

Since immediately substituting 0, for h, results in division by zero, the numerator must be simplified such that h can be factored out and then canceled against the denominator. The resulting function, f '(x), is the derivative of f(x).

Click here for some examples of how to use this quotient.

The Alternative Difference Quotient

Above, the derivative of f(x) (as defined by Newton) was described as the limit, as h approaches zero, of [f(x + h) - f(x)] / h. An alternative explanation of the derivative can be derived from Newton's quotient. Using the above; the derivative, at c, equals the limit, as h approaches zero, of [f(c + h) - f(c)] / h; if one then lets h = x - c (and c + h = x); then, x approaches c (as h approaches zero); thus, the derivative equals the limit, as x approaches c, of [f(x) - f(c)] / (x - c). This definition is used for a partial proof of the Chain Rule.

Stationary Points

Points on the graph of a function where the derivative equals zero are called "stationary points". If the second derivative is positive at a stationary point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for stationary points is often a simple way to find local minima or maxima, which can be useful in optimization.

Notable Derivatives

For logarithmic functions:
- The derivative of e^x, is e^x
- The derivative of ln x, is 1 / x.
For trigonometric functions
- The derivative of sinx, is cosx.
- The derivative of cosx, is -sinx.
- The derivative of tanx, is sec²x.
- The derivative of cotx, is -csc²x.
- The derivative of secx, is (secx)(tanx).
- The derivative of cscx, is -(cscx)(cotx).

Note that all the trigonomic "cofunctions" have negative derivatives.

Multiple Derivatives

When the derivative of a function of x has been found, the result, being also a function of x, may be also differentiated, which gives the derivative of the derivative, or second derivative. Similarly, the derivative of the second derivative is called the third derivative, and so on. One might refer to subsequent derivatives of f by:

and so on.

In order to avoid such "cumbersome" notation, the following options are often preferred:

or alternately,

Physics

Arguably the most important application of calculus, to physics, is the concept of the "time derivative" -- the rate of change over time -- which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:

Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with repsect to time) of an object's position.
Acceleration is the derivative (with respect to time) of an object's velocity.
Jerk is the derivative (with respect to time) of an object's acceleration.

Although the "time derivative" can be written "d/dt", it also has a special notation: a dot placed over the symbol of the object whose time derivative is being taken.

For example, if an object's position ; then, the object's velocity is ; the object's acceleration is ; and the object's jerk is .

If the velocity of a car is given, as a function of time; then, the derivative of said function describes the acceleration of said car, as a function of time.

Algebraic Manipulation

"Messy" limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.

Constant Rule: The derivative of any constant is zero.
- Constant Multiple Rule: If c is some real number; then, the derivative of equals c multiplied by the derivative of f(x) (a consequence of linearity below)
Linearity: for all functions f and g and all real numbers a and b.
General Power Rule (Polynomial rule): If , for some real number r; .
Product Rule: for all functions f and g.
Quotient Rule: if .
Chain Rule: If , then
Inverse functions and differentiation: If , , and f(x) and its inverse are differentiable, then for cases in which when ,
Derivative of one variable with respect to another when both are functions of a third variable: Let and . Now
Implicit differentiation: If be an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y).

In addition, the derivatives of some common functions are useful to know. See the table of derivatives.

As an example, the derivative of is .

Using Derivatives to Graph Functions

Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all "critical points" (points at which the derivative of the function has determinant zero) are mapped to local extrema; some are so-called "saddle points". The Second Derivative Test is one way to evaluate critical points: if the second derivative of the function at the critical point is positive, then the point is a local minimum; if it is negative, the point is a local maximum; if it is neither, the point is either saddle point or part of a locally flat area (possibly still a local extremum, but not absolutely so). (In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension, at local extrema.)

Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side. Also, the supremum of a continuous function on an open and bounded domain will also be one of the local maxima; the infemum will be one of the local minima--this gives one an easy way to find the bounds of the function's range.

More Info

Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Mathematicians tend to speak the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'.

The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.

In order to differentiate all continuous functions and much more, one defines the concept of distribution.

For differentiation of complex functions of a complex variable see also Holomorphic function.

References

Calculus of a Single Variable: Early Transcendental Functions (3rd Edition) by Edwards, Hostetler, and Larson (2003)

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Derivative."

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Synonyms: Derivative

Synonyms: derivative instrument (n), derived function (n), differential (n), differential coefficient (n), first derivative (n). (additional references)

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Synonyms within Context: Derivative

Context	Synonyms within Context (source: adapted from Roget's Thesaurus).
Copy	Unoriginal, imitative, derivative.
	Derivative, derivation, modification, expansion, extension, revision; second edition; (repetition).
Effect	Noun: effect, consequence; aftergrowth, aftercome; derivative, derivation; result; resultant, resultance; upshot, issue, denouement; end; development, outgrowth, fruit, crop, harvest, product, bud.
	Adjective: owing to; resulting from; Verb: derivable from; due to; caused by dependent upon; derived from, evolved from; derivative; hereditary; telegonous.
Word	Noun: word, term, vocable; name; phrase; root, etymon; derivative; part of speech; (grammar); ideophone.
	Adjective: verbal, literal; titular, nominal. conjugate, paronymous; derivative.
Source: adapted from Roget's Thesaurus.

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Crosswords: Derivative

Specialty definitions using "derivative": derivative structure ♦ material derivative, mortgage derivative ♦ partial derivative ♦ second derivative action, substantial derivative ♦ wheather derivative. (references)

Etymologies containing "derivative": Deductive ♦ Endive ♦ Importunee. (references)

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Modern Usage: Derivative

Domain	Usage
Screenplays	Too derivative. (Teenage Mutant Ninja Turtles; writing credit: Kevin Eastman; Peter Laird)
Source: compiled by the editor from various references; see credits.

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Commercial Usage: Derivative

Domain	Title
Books	Derivative Instruments Law (reference) Investments: Spot and Derivative Markets (reference) Measuring and Managing Derivative Market Risk (reference) Pricing and Hedging of Derivative Securities (reference) Pricing Derivative Securities: An Interactive, Dynamic Environment with Maple V and Matlab (reference) (more book examples)
Source: compiled by the editor from various references; see credits.

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Photo Album: Derivative

Thumbnail	Description & Credit
	Figure 20. Lucas scoop sounder, invented in 1891 by Francis Lucas of the English Telegraph Construction and Maintenance Company, this instrument was also called the "snapper." It was used primarily by ships engaged in submarine cable laying. It is a derivative of the Ross device, with elements of the British bulldog sounder.Credit: Sailing for Science - the NOAA Fleet Then and Now.
Source: compiled by the editor from various references; see credits.

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Historic Usage: Derivative

Author	Date	Quotation
Treaty of Versailles	1919	Nevertheless, the former company, if requested, shall deliver the latter company derivative copies permitting the continuation of reproduction of articles for use within German territory. (reference)
Source: compiled by the editor from various references.

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Use in Literature: Derivative

Title	Author	Quote
Les Miserables	Hugo, Victor	Man is a derivative of night.
Source: compiled by the editor from various references.

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Non-Fiction Usage: Derivative

Subject	Topic	Quote
Health		Scientists recently developed an experimental nicotine vaccine consisting of a nicotine derivative attached to a large protein. (references)
		A substance called purified protein derivative (PPD) is injected under the skin of the forearm and examined 48 to 72 hours later. (references)
		Rimantadine�a derivative of amantadine�was similarly approved in 1993. These two drugs are active against influenza A viruses but not against influenza B viruses. (references)
Economic History	Israel	There is no explicit protection for derivative works, but protection would be granted provided the derivative work includes sufficient elements of originality. (references)
	Indonesia	These limits, currently set at $3 million, apply to both each transaction and to a bank's cumulative derivative transaction position, including forward sales, swaps, and option contracts. (references)
	Bolivia	Bolivian copyright protection includes the exclusive right: to copy or reproduce work; to revise, adapt or prepare derivative works; to distribute copies of the work; and to communicate the work publicly. (references)
Political Economy	TAIWAN	The CBC still limits the use of derivative products denominated in New Taiwan Dollars (NTD). (references)
	TAIWAN	NTD-related derivative contracts may not exceed one-third of a bank's foreign exchange position. (references)
	Benin	Inflation was 3.9 percent at year's end, prompted by price hikes for imported oil and derivative products. (references)
Trade	India	Measures initiated by the Reserve Bank to integrate the Indian foreign exchange market with the global financial system include: permitting banks to fix their own position limit and aggregate gap limits; to borrow from and invest abroad up to 15 percent of tier I capital; and to arrange hedge risks for corporate clients through derivative instruments. (references)
Source: compiled by the editor from ICON Group International, Inc.; see credits.

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Usage Frequency: Derivative

"Derivative" is generally used as a noun (singular) -- approximately 69.36% of the time. "Derivative" is used about 235 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted)

Parts of Speech	Percent	Usage per 100 Million Words	Rank in English
Noun (singular)	69.36%	163	24,498
Adjective (general or positive)	30.64%	72	39,377
Total	100.00%	235	N/A

Source: compiled by the editor from several corpora; see credits.

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Expressions: Derivative

Expressions using "derivative": Derivative circulation ♦ derivative instrument ♦ derivative word ♦ first derivative ♦ hematoporphyrin derivative ♦ OTC derivative ♦ OTC derivative instrument ♦ periwinkle plant derivative ♦ second derivative action ♦ wheather derivative. Additional references.
Hypenated Usage
	Ending with "derivative": first-derivative, milk-derivative, non-derivative, period-derivative.
Source: compiled by the editor from various references; see credits.

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Frequency of Internet Keywords: Derivative

The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com.

Expression	Frequency per Day	Expression	Frequency per Day
derivative	359	derivative table	11
equity derivative	58	derivative trading	11
credit derivative	54	approximation derivative numerical taylor	11
derivative product	47	derivative math	9
calculus derivative	42	application derivative	8
weather derivative	38	derivative and hedging	8
derivative product info	36	association derivative international swap	8
derivative product training	35	derivative market	8
financial derivative	35	derivative institute.com	8
derivative product information	31	derivative job	7
accounting and derivative	27	approximation derivative difference finite	7
partial derivative	25	derivative work	7
derivative latin	23	energy derivative	7
definition derivative	16	derivative management portfolio	6
derivative institute	16	derivative first	6
derivative structuring	16	derivative citizenship	6
ammonia derivative	15	derivative empirical firm impact risk	6
derivative calculator	15	derivative protein purified	6
canadian clearing corporation derivative	14	derivative week	6
interest rate derivative	13	derivative function	6
Source: compiled by the editor from various references; see credits.

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Modern Translations: Derivative

Language		Translations for "derivative"; alternative meanings/domain in parentheses.
Albanian		i prejardhur (derived). (various references)

Arabic		‏مشتقة اللفظ, ‏مشتق, ‏المشتق, ‏ثانوى (subaltern). (various references)

Bulgarian		производна функция, производна дума, дериват. (various references)

Chinese		衍�"�物 (Derivate), 衍�"� . (various references)

Czech		derivace (derivation), derivát, odvozenina, odvozený (derived, illative), nepùvodní (unoriginal). (various references)

Danish		derivat (derivate, derivative word, derived word). (various references)

Dutch		differentiaalquotient, derivaat, derivans, financieel derivaat (derivative instrument), afleiding (abreaction, abstraction, derivate, derivation, derivative word, derived word, distraction, diversion, water abstraction), afgeleide (differentiation), afgeleid instrument (derivative instrument). (various references)

Farsi		فرعی (Accessory, Adjunct, Ancillary, By, Bye, Extraneous, Inferior, Petty, Secondary, Sideway, Subordinate, Subsidiary, Tributary), ماخوذ, مشتق (Offshoot), گرفته شده , اشتقاقی . (various references)

Finnish		johdannainen. (various references)

French		dérivé (derivate, derivative instrument, derivative word, derived, derived word, deriving). (various references)

German		ableitung (abreaction, cerebrospinal fluid shunting, conduction, CSF shunting, deduction, deflection, derivate, derivation, derivative word, derivatives, derive, derived word, differentiation, diversion, drainage, drawing off, lead, overspill, revulsion, tapping), derivat (derivative instrument), abgeleitet (deduced, derivated, derivational, derivatively, derived). (various references)

Greek		παράγωγος. (various references)

Hebrew		� �'זרת. (various references)

Hungarian		származék (progeny). (various references)

Indonesian		turunan (copy, descendant, transcript), kata jadian, bentukan (form). (various references)

Italian		derivato (derivate, derivative word, derived word, offshoot). (various references)

Japanese Kanji		デリス剤 (delimiter, delivery, Dellinger, delta, derris compound), 派�"�語 , 派�"�的 (secondary). (various references)

Japanese Katakana		デリバティブ , はせい�", はせいてき (secondary). (various references)

Korean		� 래물 (Derivate). (various references)

Manx		toarey (dung, produce, product), tayrn-sthoo, frauee (primitive). (various references)

Pig Latin		erivativeday

Portuguese		derivado. (various references)

Romanian		derivativ, derivat (derived, secondary), cuvânt derivat, acord derivat. (various references)

Russian		производное, производная, дериват. (various references)

Serbo-Croatian		izvod (excerpt, extract, precis), izvedena reč (paronym), izveden (derived). (various references)

Spanish		derivado (by product, derivable, derivate, derivative word, derived, derived word, diverted, secondary). (various references)

Swedish		avledning (derivate, derivation, derivative word, derived word, diversion, interception). (various references)

Thai		สิ่งที่พั�'นามาจากสิ่งอื่น, ลอกแบบมาจากสิ่งอื่น. (various references)

Turkish		türev (differentiation, involution), türetme (derivation, reproduction), türetilmiş şey, türetilmiş, ikincil (by-, bye-, extraneous, extrinsic, meso-, secondary, side, subsidiary). (various references)

Ukranian		вторинний, еволюційний (evolutional, evolutionary), продукт заміщення, похідне слово, похідне, похідний (marching), дериват (derivation). (various references)
Source: compiled by the editor from various translation references.

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Derivations & Misspellings: Derivative

Derivations
	Words beginning with "derivative": derivatively, derivativeness, derivativenesses, derivatives. (additional references)
	Words ending with "derivative": antiderivative, nonderivative. (additional references)
	Words containing "derivative": antiderivatives, nonderivatives. (additional references)
Misspellings
	"Derivative" is suggested in spellcheckers for the following: Derivati, derivatise, derivativ, derivativs, derivatize, derivatized, derivaty, derivitive, dirivative, drivative, reprivatize. (additional references)
Source: compiled by the editor, based on several corpora (additional references).

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Rhyming with "Derivative"

# of Phoneme Matches	Pronunciation	Word(s) rhyming with "derivative" (pronounced deri"vutiv)
5	-v u t i v	conservative, neoconservative, preservative, ultraconservative.
4	-u t i v	accusative, acquisitive, additive, affirmative, alliterative, alternative, anticompetitive, argumentative, causative, cognitive, commemorative, communicative, comparative, competitive, consecutive, consultative, contemplative, cumulative, curative, decorative, definitive, degenerative, demonstrative, diminutive, dispositive, duplicative, evocative, executive, expletive, exploitative, Federative, figurative, fixative, formative, fugitive, generative, hypersensitive, illustrative, imaginative, imperative, indicative, infinitive, informative, initiative, inoperative, inquisitive, insensitive, interpretive, intuitive, laxative, lucrative, narrative, negative, noncompetitive, noncumulative, nonexecutive, normative, nutritive, operative, palliative, participative, pejorative, positive, prerogative, preventative, primitive, prohibitive, provocative, punitive, putative, recuperative, relative, remunerative, rep, repetitive, representative, restorative, secretive, sedative, sensitive, speculative, superlative, talkative, tentative, uncompetitive, uncooperative, unimaginative, uninformative, unrepresentative, vituperative.
3	-t i v	abortive, accommodative, accumulative, active, adaptive, addictive, adjective, administrative, adoptive, affective, appointive, appreciative, assaultive, assertive, attentive, attractive, authoritative, automotive, captive, collaborative, collective, combative, conductive, congestive, connective, constructive, contraceptive, cooperative, corrective, corruptive, counterproductive, creative, deceptive, defective, deliberative, descriptive, destructive, detective, digestive, dilutive, directive, disincentive, disparages, disruptive, dissipative, distinctive, distributive, effective, elective, elucidative, eruptive, exhaustive, exploitive, facultative, festive, furtive, hyperactive, imitative, inactive, inattentive, incentive, ineffective, infective, injunctive, innovative, instinctive, instructive, interactive, introspective, invective, inventive, investigative, irrespective, iterative, legislative, locomotive, manipulative, meditative, motive, native, nonautomotive, nonnative, nonproductive, objective, obstructive, octave, overactive, perceptive, perspective, photoconductive, plaintive, predictive, preemptive, presumptive, preventive, proactive, probative, productive, prognosticative, projective, prospective, protective, qualitative, quantitative, radioactive, reactive, receptive, reconstructive, redemptive, redistributive, reflective, refractive, regulative, rehabilitative, reproductive, respective, restive, restrictive, retroactive, retrospective, seductive, selective, stimulative, subjective, substantive, suggestive, superconductive, supportive, unattractive, unproductive, unreceptive, vegetative, vindictive.
Source: compiled by the editor (additional references); see credits.

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Anagrams: Derivative

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "a-d-e-e-i-i-r-t-v-v"
	-2 letters: derivate, ideative.
	-3 letters: averted, deviate, revived, riveted, vetiver, vivider.
	-4 letters: advert, aeried, airted, dative, dearie, derate, derive, dieter, divert, evader, evited, ideate, reaved, redate, rediae, reedit, reived, retied, revive, revved, teared, tidier, tiered, tirade, trivia, varied, varved, verite, vervet.
	-5 letters: aerie, aider, aired, aiver, arete, avert, dater, davit, deair, deave, derat, deter, diver, drave.
	Words containing the letters "a-d-e-e-i-i-r-t-v-v"
	+1 letter: derivatives.
	+2 letters: derivatively.
	+3 letters: nonderivative.
	+4 letters: antiderivative, derivativeness, nonderivatives.
	+5 letters: antiderivatives, overadvertising.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.